Meshless Methods Introduction - Meshless Methods in Biomechanics - page 73

Fig. 3.19 Representation of
the essential boundaries

Using the nodal sets X Cu ð C Þ and X Cu ð S Þ , two new sets of integration points
Q Cu ð C Þ ¼ q 1 ; q 2 ; ... ; q QC

n o 2 C u ð C Þ and Q Cu ð S Þ ¼ f q 1 ; q 2 ; ... ; q QS g2 C u ð S Þ are
respectively defined, in which QC represents the number of integration points
along the boundary curve C u ð C Þ and QS stands for the number of integration point
inside of the area of the essential boundary surface C u ð S Þ . As mentioned in
Sect. 3.4.4 , each integration point on set Q Cu ð C Þ represents an infinitesimal curve
and the integration points on set Q Cu ð S Þ represent infinitesimal surface areas.
Therefore, the integration weight _ C

I of each integration point represents a length if
q I 2 Q Cu ð C Þ , or a surface area if q I 2 Q Cu ð S Þ . In Fig. 3.19 are presented the field
nodes and the integration points along the essential boundary curve C u ð C Þ and the
essential boundary surface C u ð S Þ .
Additionally, it is required to establish new influence-domains for each one
interest point q I discretizing the essential boundary. However, the nodes making
the new influence-domains have to belong to the essential boundary curve C u ð C Þ if
q I 2 Q Cu ð C Þ , or to the essential boundary surface C u ð S Þ if q I 2 Q Cu ð S Þ . Notice that
the meshless shape functions constructed to approximate the field variable on the
boundary for each one integration point q I are constructed using only the field

nodes on the essential boundary.
Neglecting the dynamic term of the Galerkin weak form presented in Eq. ( 2.65 )
and adding the Lagrange multipliers, the Galerkin weak form expression can be
written as,

2
4

3
5 ¼ 0

Z

d L ð T cL ð dX Z
X

du T b dX Z
Ct

Z

du T t dC t d k T ð u u Þ dC u

X Cu

ð 3 : 38 Þ

The last variational term results from the Lagrange multipliers method and it is
included in the expression to ensure: u u ¼ 0. The last term can be developed,